   Chapter 11.2, Problem 105E

Chapter
Section
Textbook Problem

# GeometryWrite an equation whose graph consists of the set of points P ( x , y , z ) that are twice as far from A ( 0 , − 1 , 1 ) as from B ( 1 , 2 , 0 ) . Describe the geometric figure represented by the equation.

To determine
The equation where the graph consists of the point given as, P(x,y,z), where A(0,1,1),B(1,2,0) and PA=2PB.

Explanation

Given:

The points are A(0,1,1),B(1,2,0) and PA=2PB

Explanation:

Consider, A(0,1,1),B(1,2,0) and P(x,y,z)

PA=(x0)2+(y+1)2+(z1)2=x2+(y+1)2+(z1)2PB=(x1)2+(y2)2+(z0)2=(x1)2+(y2)2+z2

Since, PA=2PB

x2+(y+1)2+(z1)2=2(x1)2+(y2)2+z2

Squaring both sides

x2+(y+1)2+(z1)2=4[(x1)2+(y2)2+z2]x2+y2+2y+1+z22z+1=4[x22x+1+y24y+4+z2]x2+y2+z2+2y2z+2=4x2+4y2+4z28x16y+204x2x2+4y2y2+4z2z28x18y+2z+18=0

Now,

3x2+3y2+3z28x18y+2z+18=0x2+y2+z283x6y+23z+6=0x22

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